Our bound relies on a known result that relates the uniform mixing time of the lazy random walk on a graph to the size of “bottlenecks” [1]}, i.e. small sets that separate the graph into two large connected components.
The narrower the bottlenecks, the harder it is for the random walk to cross them, and the longer the mixing time. Conversely, if there is no very narrow bottleneck then the mixing time will not be too large.
The bulk of this article is thus dedicated to showing that bottlenecks cannot be too narrow, see Theorem REF  and Corollary REF .
